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By Ferreira, Lopez, Mainar.

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F(a). It is the obtained property combined with the positiveness of I that ensures the sought estimates. We now have to establish that the functional constructed is invariant under shifts. • that 1(, a) -l(a) for all a EICXJ. And again. the sequence a can be considered standard. in which case the element 'a is also standard and. hence. Here account has been made of the finiteness of the numbers a2N / N and aN / N. 18. 8. Nonstandard criterion for continuity. Let f be a standard real junction and x be a standard point of its standard domain of definition, dom (j).

Obviously, A .. 0 and Asn, where the standard number nEoN is such that -ns ts n. Indeed, for every standard a EA we have as ts n. By the transfer principle, we deduce As n. Since R is complete, we have s:= supAER. Obviously, s is a standard number. Let us show that s ... t . In the opposite case we get, for a certain standard > 0, Is - tl> £. , s ~ a + £. In this case, however, we would get s ~ s + e , which is impossible. The remaining possibility, s < t also results in a contradiction, since then we would get t > S + e, and, again, s ~ s + e .

And objects like .... "'R. 0. etc.. Still more unpleasant is the fact that the 'mathematical law' x -+ ox. acting from R to R. is not a function. The point is that the notion of function had been formed in mathematics long before the set-theoretic stance was developed. Thus. Euler wrote: "When certain quantities depend on other ones in such a way that when the latter are changed the former are also subject to changes. then the former are called functions of the latter. The application of this term is extremely wide; it embraces all the methods by which one quantity can be determined with the help of others.

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